Given that 1D space $\mathbb{R}$ and 3D space $\mathbb{R}^3$ are in bijection, why do we describe our physical world as 3D? Mostly the reason given is that that three numbers are required to specify a point uniquely in our world. But this is utter nonsense!
It has long been mathematically proven that $\mathbb{R}$ is in bijection with $\mathbb{R}^3$, i.e. $\mathbb{R}$ and $\mathbb{R}^3$ have the same cardinality. So what is the “additional theoretical structure” (which has to be physical, not mathematical, since we’re describing our physical world) that prompts us to label our space as “3D”?

Note: I am not talking about including time as the fourth dimension. This is under the theory of relativity. And I am asking a more general question.
 A: The key is to view space(time) $M$ as a differential manifold with a differential structure rather than just a set. One may show that two differential manifolds $M$ and $N$ can only be diffeomorphic if they have the same dimension.
A: The problem with such bijections, also known as space-filling curves, is that they are cumbersome to work with, and have some properties that make them useless for physical theories. For example, (square of) distance between two points in 3D Euclidean space is a smooth function:
$$d^2=\Delta x^2+\Delta y^2+\Delta z^2.$$
But in a space-filling curve this would be a discontinuous function of the two curve parameters corresponding to the two points. This then leads to impossibility of convenient description of trajectories, and this affects the whole of physics, starting already from kinematics.
A: $\mathbb{R}$ is in set-theoretic bijection with $\mathbb{R}^3$ but the two are not homeomorphic. That means that to a mathematician they are only 'the same' as sets, and not as topological spaces. In particular, any bijection between them must be discontinuous in at least one direction. For instance, space filling curves do not have continous inverses.
The topological structure of space is incredibly important. Physicists do a lot more than just talk about abstract sets, we have a notion of closeness that topology furnishes. Of course, many theories have even more structure than this. You might be able to actually measure the distance between points for instance! Regardless, we normally need some mathematical structure beyond set theory for the theory to make any physical sense. You might as well ask why we don't describe the universe as a circle, since they are also in bijection with $\mathbb{R}^3$ - the reason is that any structure that meaningfully makes it 'a circle' includes the (topological) fact that you can wrap around on yourself, which is not observed in our space. 
